Generalized Laguerre Quasi-Orthogonal Approximations And Petrov-Galerkin Domain Decomposition Spectral Method For High Order Problems With Mixed Inhomogeneous Boundary Conditions

As an important numerical method for solving differential equations, spectral method devel-oped rapidly during the past three decades. It has been used widely for many problems in science and engineering. The fascinating merit of spectral method is its high accuracy. The early spectral method is only available for periodic problems and some problems defined on bounded rectan-gular domains. However, in many practical cases, we need numerical algorithms for unbounded domains, with high accuracy.The purpose of this work is to investigate three kinds of new generalized Laguerre orthogo-nal projection, quasi-orthogonal projection and the related interpolation. We establish the basic results. Some of them are optimal. Accordingly, we propose Petrov-Galerkin spectral method and domain decomposition spectral method for differential equations of high order.We first introduce the generalized Laguerre functions L?(?,?)(x). They are mutually orthogo-nal with the weight x?e-?x, where?is any real number and?>0. We establish the basic results on the corresponding orthogonal approximation, quasi-orthogonal approximation and Laguerre-Gauss-Radau interpolation. These results lead to new Petrov-Galerkin spectral and collation meth-ods.Next, we introduce the generalized Laguerre functions L?(?,?)(x) which are mutually orthogo-nal with the weight x?,?being any real number. The basic results on the corresponding orthogo-nal approximation, quasi-orthogonal approximation and Laguerre-Gauss-Radau interpolation are established. They form the mathematical foundation of new Petrov-Galerkin domain decompo-sition spectral method for differential equations of high order defined on unbounded domains, as well as certain exterior problems.Finally, for fitting the asymptotic behaviors of exact solutions at infinity properly, we in-troduce the Laguerre orthogonal approximation, quasi-orthogonal approximation and Laguerre-Gauss-Radau interpolation, by taking the generalized Laguerre functions Ll(?,?,?,?)(x) as the basis functions, which are mutually orthogonal with the weight function x?(?+x)-?,?> 0,?and?being any real numbers. Accordingly, we provide new spectral method, which matches the asymptotic behaviors of exact solutions closely.Numerical results demonstrate the high accuracy of the suggested new spectral methods, and coincide with the analysis very well.